IGNOU PHE 5 2018 - Reduce the following PDE to a set of three ODEs by the method of separ

Question

Reduce the following PDE to a set of three ODEs by the method of separation of
variables

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27 Jan 2018

Answer :

Word Count : 437

We are tasked with reducing the following partial differential equation (PDE) to a set of three ordinary differential equations (ODEs) using the method of separation of variables: \[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} \] This is the heat equation, where: - \( u(x, t) \) is the temperature distribution as a function of position \( x \) and time \( t \), - \( \alpha \) is the thermal diffusivity (a constant). ### Step 1: Assume a Separable Solution We assume that the solution \( u(x, t) \) can be written as the product of two functions, one depending only on \( x \) and the other depending only on \( t \): \[ u(x, t) = X(x)T(t) \] ### Step 2: Substitute into the PDE Substitute \( u(x, t) = X(x)T(t) \) into the heat equation: \[ \frac{\partial}{\partial t} [X(x)T(t)] = \alpha \frac{\partial^2}{\partial x^2} [X(x)T(t)] \] Simplify the derivatives: \[ X(x) \frac{dT}{dt} = \alpha T(t) \frac{d^2X}{dx^2} \] ###

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